of 1 16
The Human Scale as a Resonant Node Between the Quantum and
Astrophysical Scales
Ian Beardsley
September 30, 2026
Abstract
We propose that the human scale — the characteristic mass, length, and energy of biological
organisms — is not an arbitrary biological fact but a resonant node between the quantum scale
and the Earth–Moon–Sun scale. Using the Universal Particle Law (UPL), which posits a one-
second resonance governing bound systems from the proton to the galaxy, we construct a human
effective action , where is Planck's constant and is the
Earth–Moon–Sun effective action. With the UPL's one-second timescale, this gives a human
energy scale — on the order of the mechanical work of a heartbeat — and a
characteristic length-to-mass ratio (for ). The geometric means of
particle masses and astrophysical masses bracket the human mass range: ,
, , and . Extending the cascade to the
galactic scale, we identify two additional resonant nodes — the asteroid belt at ~2.4 AU and
Saturn at ~9.5 AU — that complete a scale hierarchy to . We suggest this
hierarchy provides a quantitative framework for astrobiology: the conditions for human-scale
intelligence may be geometrically fixed, and the search for life should target systems with the
same resonant architecture.
1. Introduction
The Universal Particle Law (UPL) proposes that inertia is the geometric resistance to rotating a
particle's velocity vector out of the time dimension and into space [1]. The normal force resisting
this rotation is
where is Planck's constant, is the speed of light, and is a fundamental timescale.
When applied to the proton, neutron, and electron, the law returns :
h
human
= h h
⊙
≈ 3.4248 J·s
h
h
⊙
E
human
≈ 3.4248 J
r /m ≈ 0.0431 m/kg
κ
human
= 1
m
e
M
⊕
≈ 2.3 g
m
e
M
⊙
≈ 1.35 kg
m
p
M
⊙
≈ 57.7 kg
h → h
human
→ h
⊙
h
gal
F
n
=
h
ct
2
1
,
h
c
t
1
≈ 1 s
t
1
≈ 1 s
t
1
=
r
i
m
i
πh
Gc
κ
i
,
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with and . The same law, applied to the Earth–Moon–Sun
system with an effective action , returns . Applied to the
Sun's orbit around the galactic center with , it returns at two
geometric nodes: the asteroid belt (~2.4 AU) and Saturn (~9.5 AU).
A natural question arises: where does the human scale fit? Is there an effective action that,
when inserted into the UPL with , returns human-scale masses and lengths? In this paper,
we show that the geometric mean of the quantum and solar effective actions,
does exactly this. We then extend the cascade to the galactic scale and identify the astrobiology
implications.
2. The Quantum and Solar Effective Actions
2.1 The quantum scale
The UPL at the particle level uses the fundamental Planck constant:
The law is satisfied for the electron, proton, and neutron:
Particle Properties!
The particle scale is characterized by and .
2.2 The Earth–Moon–Sun scale
For the Earth–Moon–Sun system, the effective action is computed from Earth's orbital kinetic
energy around the Sun:
The celestial Planck-type constant is defined as
κ
p
= κ
n
= 1/(3α
2
) ≈ 6256.33
κ
e
= 1
h
⊙
≈ 1.77018 × 10
34
J·s
t
1
≈ 0.982 s
h
gal
≈ 3.0243 × 10
41
J·s
t
1
≈ 1 s
h
human
t
1
= 1 s
h
human
= h h
⊙
,
h = 6.62607015 × 10
−34
J·s .
Particle
rᵢ (m)
mᵢ (kg)
κᵢ
t₁ (s)
Electron
2.81794 × 10⁻¹⁵
9.10938 × 10⁻³¹
1
0.99773
Proton
0.833 × 10⁻¹⁵
1.67262 × 10⁻²⁷
1/(3α²)
1.00500
Neutron
0.8367 × 10⁻¹⁵
1.675 × 10⁻²⁷
1/(3α²)
1.00803
h
t
1
≈ 1 s
K E
⊕
=
1
2
(5.972 × 10
24
kg)(29,785 m/s)
2
≈ 2.649 × 10
33
J .
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The UPL at this scale takes the form
where , , and the geometric coupling is the eclipse
ratio
This returns
The Earth–Moon–Sun scale is characterized by and the same one-second resonance.
3. The Human Effective Action
3.1 Definition
The quantum scale is characterized by . The solar scale is characterized by
. The geometric mean is
This is the natural intermediate action between the quantum and astrophysical scales.
3.2 The human energy scale
With the UPL's one-second resonance, , the human energy scale is
This is on the order of everyday human mechanical work:
h
⊙
= 2π (1 s)K E
⊕
≈ 1.77018 × 10
34
J·s .
t
1
=
R
⊕
M
⊕
πh
⊙
Gc
κ
⊕
,
R
⊕
= 6.371 × 10
6
m
M
⊕
= 5.972 × 10
24
kg
κ
⊕
=
r
moon
R
⊙
=
3.84399 × 10
8
6.96 × 10
8
≈ 0.5522974.
t
1
= 1.7785 s × 0.5522974 ≈ 0.982 s ≈ 1 s .
h
⊙
h ∼ 10
−34
J·s
h
⊙
∼ 10
34
J·s
h
human
= h h
⊙
= (6.62607015 × 10
−34
)(1.77018 × 10
34
) ≈ 3.4248 J·s .
t
1
= 1 s
E
human
=
h
human
t
1
≈ 3.4248 J .
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- Lifting a 1 kg mass through 35 cm (On the order of 2 lbs through 1 ft in Earth gravity):
- Lifting a 0.1 kg mass through 3.5 m:
The human heartbeat does approximately 1 J of mechanical work per beat; at ~1 Hz, this
corresponds to a power of ~1 W. The energy scale is on the order of the
mechanical work of a few heartbeats, or a single arm movement.
3.3 The human length-to-mass ratio
Applying the UPL at the human scale:
Taking and :
This gives a characteristic length for any human-scale mass:
Mass Scale and Predicted Radius (κ = 1)!
A 70–100 kg human corresponds to a characteristic length of 3–4 m. This is the human scale —
not the body itself, but the space a human occupies, the reach, the stride, the arm span.
If we instead use (The eclipse ratio for the Moon near perfectly eclipsing
the Sun as seen from the Earth), the length scales shrink by about half, bringing 100 kg to about
2.4 m — closer to human height. The exact value of is a choice; the order of magnitude is
robust.
mgh = (1)(9.8)(0.35) ≈ 3.43 J .
(0.1)(9.8)(3.5) ≈ 3.43 J .
E
human
≈ 3.4 J
t
1
=
r
human
m
human
πh
human
Gc
κ
human
.
κ
human
= 1
t
1
= 1 s
r
human
m
human
=
Gc
πh
human
≈ 0.0431 m/kg .
Mass
Predicted r (κ = 1)
Human analogue
1 g
0.043 mm
A grain of sand
100 g
4.3 mm
A pea
1 kg
4.3 cm
A small fruit
10 kg
43 cm
A torso
70 kg
3.0 m
Arm span
100 kg
4.3 m
A tall stride
κ
human
= κ
⊕
≈ 0.552
κ
human
of 5 16
4. Human-Scale Masses from Geometric Means
The geometric means of particle masses and astrophysical masses bracket the human mass range:
Geometric Mean Masses!
The proton–Earth and proton–Sun geometric means are exactly human-scale masses. The
electron–Earth and electron–Sun means are also human-scale, though smaller. The human body
sits at the geometric midpoint between the quantum and astrophysical scales.
This is a striking quantitative result: the human mass range is the geometric mean of the quantum
and astrophysical mass scales.
5. The Full Cascade
5.1 The scale hierarchy
The human scale is the middle rung of a cascade:
Each level satisfies the same one-second law with a different effective action:
Effective Action Across Scales!
5.2 The galactic nodes
From the first paper, the galactic level has two geometric nodes:
Pair
Geometric mean mass
Human equivalent
mₑ and M⊕
2.33 × 10⁻³ kg = 2.3 g
A portion of food
mₚ and M⊕
0.100 kg = 100 g
A small meal
mₑ and M☉
1.35 kg
A newborn
mₚ and M☉
57.7 kg
A six-foot man
h ⟶ h
human
⟶ h
⊙
⟶ h
gal
.
Level
Effective action
Characteristic scale
t₁
Quantum
h = 6.626 × 10⁻³⁴ J·s
Proton, neutron, electron
~1 s
Human
h_human = 3.4248 J·s
1 g – 100 kg
~1 s
Earth–Moon–
Sun
h☉ = 1.77018 × 10³⁴ J·s
Earth orbit, Moon, eclipse
~1 s
Galaxy
h_gal = 3.0243 × 10⁴¹ J·s
Asteroid belt, Saturn
~1 s
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- Asteroid belt at (~2.4 AU) — the region that delivered water and organics to the
early Earth.
- Saturn at (~9.5 AU) — a gravitational shield deflecting comets and asteroids.
These are resonant "sweet spots" of the galactic potential, fixed by the Sun's mass, radius, and
motion through the galaxy.
5.3 The scaling operator
The same operator relates all levels:
This operator takes , , and . It contains no free parameters; the
effective actions are computed independently from kinetic energies.
6. Astrobiology Implications
6.1 The resonant architecture
If the human scale is a resonant node between the quantum and astrophysical scales, and if the
Earth–Moon–Sun system is tuned to produce this node, then life requires a planetary system with
specific architectural features:
1. A rocky planet at ~1 AU — where liquid water is stable.
2. An asteroid belt at ~2.4 AU — to deliver water and organics.
3. A gas giant at ~9.5 AU — to shield the inner system.
4. A large moon — to stabilize obliquity and produce a ~24-hour day.
5. A perfect eclipse geometry — , the eclipse ratio.
The UPL predicts these from the Sun's mass, radius, and galactic motion. They are not
independent coincidences; they are resonant nodes of the same one-second law.
6.2 The search strategy
The astrobiology prediction is specific and falsifiable:
Look for exoplanetary systems with an asteroid belt at ~2.4 AU and a gas giant at ~9.5 AU.
These are the systems where the human-scale resonance can emerge.
κ
ast
gal
≈ 0.415
κ
sat
gal
≈ 0.115
h
b
h
a
=
(
R
a
M
b
κ
a
M
a
R
b
κ
b
)
2
.
h → h
human
h
human
→ h
⊙
h
⊙
→ h
gal
r
m
/R
⊙
≈ 0.55
of 7 16
This is a concrete target. Current and future missions (JWST, PLATO, HabEx, LUVOIR) can
characterize exoplanetary system architecture. The prediction is that systems with the full
resonant architecture are more likely to host complex life than systems without it.
6.3 Why this is not anthropocentric
The human scale is not put in by hand. It emerges from the geometric mean of the quantum and
solar effective actions:
The Earth–Moon–Sun system is tuned to produce this mean at the one-second resonance. If this
is correct, then the conditions for human-scale intelligence are not arbitrary — they are
geometrically fixed. This gives us a concrete target for astrobiology.
7. Discussion
7.1 What is established
Three things can be said with confidence:
1. The human effective action exists. is the natural geometric
mean between the quantum and solar scales.
2. The human energy scale is human. is on the order of everyday human
mechanical work.
3. The human mass range is bracketed by geometric means. to
.
7.2 What is not established
The UPL does not derive the value of from first principles. It does not derive the
couplings . The geometric means are numerical observations within the UPL framework, not
derivations from a deeper theory. The astrobiology prediction is a hypothesis to be tested.
The author finds the following holds for the kinetic energies of the moon and the Earth:
h
human
= h h
⊙
.
h
human
= h h
⊙
≈ 3.4248 J·s
E
human
≈ 3.4248 J
m
e
M
⊕
≈ 2.3 g
m
p
M
⊙
≈ 57.7 kg
t
1
= 1 s
κ
i
K E
m
K E
e
(Ear th Day) = 1.25seconds
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We used average orbital velocities. And, accounting for the Earth’s inclination to its orbit ( =
23.5 deg), we have
Earth day=(24)(60)(60)=86,400 seconds. We can bring down equations these equations closer to
a second using the Moon’s orbital velocity at aphelion, and Earth’s orbital velocity at perihelion.
We have:
The Moon allows for the evolution of complex, intelligent life because it stabilizes the earth tilt
to its orbit allowing for the seasons and preventing temperature extremes. We live in an
interesting time when the Moon near perfectly eclipses the Sun. This is because while the Sun is
400 times larger than the Moon, it is 400 times further from the Earth than the Moon is. That is
the perfect eclipse is given by the orbital radius of the Earth to the Moon’s orbital radius
equals the Solar radius to the lunar radius :
The Earth's rotation is slowing down, and as a direct result, the Moon is slowly drifting away.
Earth's Rotation Slowing: The length of a day is increasing by about 2.3 milliseconds per
century. This is the long-term average rate caused primarily by the Moon's tidal pull. (Note that
this rate can be influenced by other factors; for example, climate change is currently contributing
about 1.33 milliseconds per century).
Moon's Orbit Growing: The Moon is receding from Earth at a rate of about 3.8 centimeters (1.5
inches) per year.
K E
m
=
1
2
(7.4767E 22kg)(1,022m /s)
2
= 3.83726E 28J
K E
e
1
2
(5.972E 24kg)(29,785m /s)
2
= 2.649E 33J
θ
e
K E
m
K E
e
(Ear th Day)cos(θ
e
) = 1.146seconds
K E
m
=
1
2
(7.347673E 22kg)(966m /s)
2
= 3.428E 28J
K E
e
=
1
2
(5.972E 24kg)(30,290m /s)
2
= 2.7396E 33J
K E
m
K E
e
(Ear th Day)cos(θ
e
) = 0.991seconds
K E
m
K E
e
(Ear th Day)cos(θ
e
) ≈ 1second
r
e
r
m
R
⊙
R
m
r
e
r
m
=
R
⊙
R
m
of 9 16
The Tidal Connection
These two phenomena are linked by tidal friction. The Moon's gravity creates tides in Earth's
oceans. As Earth rotates, the tidal bulges are pulled slightly ahead of the Moon, creating friction
that slows Earth's spin.
To conserve angular momentum in the Earth-Moon system, this loss of Earth's rotational speed is
transferred to the Moon, boosting its orbital energy and causing it to spiral outward into a larger
orbit. This recession rate has been precisely measured for decades using the Lunar Laser Ranging
experiment, which bounces lasers off reflectors left on the Moon by Apollo astronauts.
Long-Term Perspective
While these changes are tiny on human timescales, they have significant effects over millions of
years. About 250 million years ago, a day on Earth was roughly 23 hours long. In about 600
million years, the Moon will have drifted so far that total solar eclipses will no longer be
possible.
We might guess from the ground state of the Bohr hydrogen atom, which is given by
that a Planck-type constant is given by the Moon, if it is the metric by which the Solar System
does its measuring, that it is given by:
Where is a celestial Planck type constant for the Earth/Moon/Sun system, is the mass of
the Moon, and we have let: , , . This gives us that:
Nicely, this has that:
Suggesting that that the quantum analog for the Earth/Moon/Sun system and the Universal
Particle Equation both have a characteristic time of about 1 second.
7.3 Selection effects
We observe the human scale because we are human. This is a selection effect that cannot be ruled
out without a broader statistical argument. However, the fact that the human scale emerges from
a
0
=
ℏ
2
k
e
m
e
e
2
ℏ
2
⊙
GM
3
m
1
c
= 1.00seconds
ℏ
⊙
M
m
ℏ → ℏ
⊙
k
e
→ G
m
e
e
2
→ M
3
m
ℏ
⊙
= 2.81733E 33J ⋅ s
h
⊙
= 2π ℏ
⊙
= 1.77018E 34J ⋅ s
ℏ
⊙
≈ (1.00seconds)K E
e
of 10 16
the geometric mean of the quantum and solar actions is not a selection effect; it is an algebraic
fact about the UPL.
7.4 Testability
The framework makes specific predictions:
- The UPL should hold at the human scale with and .
- Exoplanetary systems with an asteroid belt at ~2.4 AU and a gas giant at ~9.5 AU should be
more likely to host complex life.
- Systems without these features should be less likely to host complex life.
These predictions are testable with current and future data.
8. Conclusion
We have shown that the human scale emerges as a resonant node between the quantum scale and
the Earth–Moon–Sun scale. The human effective action is the geometric mean of the Planck
constant and the celestial effective action:
With the UPL's one-second resonance, this gives a human energy scale and a
characteristic length-to-mass ratio . The geometric means of particle and
astrophysical masses bracket the human mass range. Extending the cascade to the galactic scale,
we identify the asteroid belt and Saturn as additional resonant nodes.
The human scale is not an arbitrary biological fact. It is the middle rung of a scale hierarchy that
spans from the proton to the galaxy. This suggests a quantitative framework for astrobiology: the
conditions for human-scale intelligence may be geometrically fixed, and the search for life
should target systems with the same resonant architecture.
References
[1] Beardsley, I. (2026). Scale Invariance and Fractal Features in Cosmology: A Computational
Study. Zenodo. https://doi.org/10.5281/zenodo.23045265
[2] Beardsley, I. (2026). The Human Scale and Natural Laws.
[3] Tiesinga, E., Mohr, P. J., Newell, D. B., & Taylor, B. N. (2024). CODATA Recommended
Values of the Fundamental Physical Constants. NIST.
h
human
κ
human
≈ 1
h
human
= h h
⊙
≈ 3.4248 J·s .
E
human
≈ 3.4248 J
r /m ≈ 0.0431 m/kg
of 11 16
[4] Chiaramonte, F., & Loeffen, R. (2026). VGT-CIT Reduction and the Status of the UPL as a
Limiting Case. Private communication.
[5] Tynski, K. (2024). One Equation, ~200 Mysteries: A Structural Constraint That May Explain
(Almost) Everything.
Appendix: Numerical Values
Physical Quantities!
Appendix: Dimensionless Reformulation of the UPL Hierarchy
A.1 Motivation
A natural objection to the Universal Particle Law (UPL) is that its central result, second,
appears to depend on the human convention of the second. The second, kilogram, meter, and
joule are human units. If the UPL describes a structural resonance, its significance should survive
the removal of human measurement conventions.
This appendix addresses that objection directly. We reformulate the UPL in an -normalized
natural unit system, show that the law collapses to a pure ratio statement, and demonstrate that
the full hierarchy — electron, proton, neutron, Earth–Moon–Sun, human, galaxy — converges to
a single dimensionless coordinate. The resonance survives the removal of SI units.
Quantity
Value
h
6.62607015 × 10⁻³⁴ J·s
h☉
1.77018 × 10³⁴ J·s
h_human
3.4248 J·s
E_human
3.4248 J
r/m (κ = 1)
0.0431 m/kg
√(m_e M_⊕)
2.33 g
√(m_p M_⊕)
100 g
√(m_e M_☉)
1.35 kg
√(m_p M_☉)
57.7 kg
h_gal
3.0243 × 10⁴¹ J·s
κ_gal^ast
0.415
κ_gal^sat
0.115
t
1
≈ 1
h
of 12 16
The reformulation was suggested by Volynets Evgeny Vatslavovich (author of MASV-Prime) in
a comment on the paper under discussions at academia.com. We present it here because it
strengthens the framework and clarifies what is structural versus what is metrological.
A.2 The -normalized natural unit system
Conventional Planck units use . Because the UPL uses rather than , we construct a
natural unit system from , , and :
These are not the standard Planck units. They are the natural units that speak the same
mathematical language as the UPL.
In these units, one SI second corresponds to
This is the dimensionless content of "one second." It does not depend on the caesium atom, the
division of the day, or any human convention. It is a pure ratio.
A.3 The UPL in natural units
The UPL is
Substituting , , and :
h
ℏ = h /(2π)
h
ℏ
h
c
G
t
h
=
hG
c
5
= 1.351385078 × 10
−43
s,
ℓ
h
=
hG
c
3
= 4.051350543 × 10
−35
m,
m
h
=
hc
G
= 5.455511861 × 10
−8
kg,
E
h
= m
h
c
2
=
hc
5
G
= 4.903169538 × 10
9
J .
1 s
t
h
= 7.3998 × 10
42
.
t
1
=
r
i
m
i
πh
Gc
κ
i
.
r
i
= R
i
ℓ
h
m
i
= M
i
m
h
t
1
= T
i
t
h
of 13 16
Using and , this collapses to
The UPL becomes a pure ratio law. No dimensionful constants remain. The resonance condition
second becomes
This is the structural statement. The value second is its human projection.
A.4 The full hierarchy in natural units
We evaluate at each level of the hierarchy, using the same inputs as the main
text.
A.4.1 Particle level
Particle Data!
A.4.2 Earth–Moon–Sun level
A.4.3 Human level
For a 70 kg human with a characteristic length of 3 m (arm span, stride, or occupied space):
T
i
t
h
=
R
i
ℓ
h
M
i
m
h
πh
Gc
κ
i
.
ℓ
h
/m
h
= G /c
2
t
h
= hG /c
5
T
i
= π κ
i
R
i
M
i
.
t
1
≈ 1
T
i
≈ 7.4 × 10
42
.
1
T
i
= π κ
i
R
i
/M
i
Particle
rᵢ (m)
mᵢ (kg)
κᵢ
Tᵢ
Electron
2.81794 × 10⁻¹⁵
9.10938 × 10⁻³¹
1
7.382 × 10⁴²
Proton
0.833 × 10⁻¹⁵
1.67262 × 10⁻²⁷
6256.33
7.436 × 10⁴²
Neutron
0.8367 × 10⁻¹⁵
1.675 × 10⁻²⁷
6256.33
7.459 × 10⁴²
R
⊕
= 6.371 × 10
6
m, M
⊕
= 5.972 × 10
24
kg, κ
⊕
= 0.5522974.
T
⊕
= π (0.5522974)
R
⊕
M
⊕
= 7.27 × 10
42
.
R
human
= 3 m, M
human
= 70 kg, κ
human
= 1.
of 14 16
A.4.4 Galactic level
For the asteroid node:
For the Saturn node:
A.4.5 Summary table
Scale and Tᵢ Values!
Every scale converges to . The spread is less than 3%.
A.5 What this establishes
Three things can be said with confidence.
First, the resonance is structural, not metrological. The convergence to does not
depend on seconds, kilograms, or meters. It is a dimensionless statement about the ratio
weighted by the coupling . The UPL passes the stress test of removing human units.
T
human
= π (1)
3
70
= 7.35 × 10
42
.
R
⊙
= 6.96 × 10
8
m, M
⊙
= 1.989 × 10
30
kg, κ
ast
gal
= 0.415.
T
ast
gal
= π (0.415)
R
⊙
M
⊙
= 7.28 × 10
42
.
r
⊙
= 2.5 × 10
20
m, M
gal
= 1.989 × 10
41
kg, κ
sat
gal
= 0.115.
T
sat
gal
= π (0.115)
r
⊙
M
gal
= 7.28 × 10
42
.
Scale
Tᵢ
Electron
7.382 × 10⁴²
Proton
7.436 × 10⁴²
Neutron
7.459 × 10⁴²
Human (70 kg, 3 m)
7.35 × 10⁴²
Earth–Moon–Sun
7.27 × 10⁴²
Galaxy (asteroid)
7.28 × 10⁴²
Galaxy (Saturn)
7.28 × 10⁴²
One SI second
7.3998 × 10⁴²
T ≈ 7.3-7.5 × 10
42
T ≈ 7.4 × 10
42
R /M
κ
of 15 16
Second, the human scale is on the resonance. The human entry is as
close to the one-second value as the proton or electron. The human scale is not an outlier; it sits
on the same resonant node.
Third, SI units are appropriate precisely because the resonance is human-scale. We use seconds
and joules not because we are confusing the map for the territory, but because the resonant node
is the human scale. The dimensionless ratio maps to 1 SI second because the human
scale is where the resonance lives. If the resonance were at the Planck scale, we would use
Planck units. If it were at the galactic scale, we would use kiloparsecs and solar masses. We use
SI because the human scale is the node.
A.6 What this does not establish
The dimensionless reformulation does not derive the value from first principles. It
does not explain why the couplings take the values they do. It does not derive the one-second
scale from a deeper theory.
What it does show is that the resonance is not an artifact of human measurement conventions.
The same dimensionless coordinate appears at every level of the hierarchy, from the electron to
the galaxy. That is a structural fact about the UPL, not a metrological accident.
A.7 The two views are the same fact
Views of the Resonance Time Scale!
All three are correct. They are not competing claims. The dimensional view is the human
projection of the dimensionless fact. The dimensionless view is the structural fact that projects
into human units.
The territory happens to include the mapmaker.
A.8 A note on the golden-ratio coordinate
Volynets also suggested a logarithmic coordinate with the golden ratio at the origin:
T
human
≈ 7.35 × 10
42
7.4 × 10
42
7.4 × 10
42
κ
i
View
Statement
What it emphasizes
Dimensional (SI)
t₁ ≈ 1 second
The human scale is the
resonance
Dimensionless
(natural)
t₁/tₕ ≈ 7.4 × 10⁴²
The resonance is structural
Ratio law
T = √π κ R/M
The mechanism is geometric
of 16 16
This is a coordinate convention. Shifting the origin of a logarithmic coordinate does not create
new physics. The physically meaningful quantity is the persistence of the same dimensionless
relations after changing coordinates and removing human units. The golden-ratio origin is a
convenience, not a physical claim. We mention it here only to acknowledge the suggestion and to
distinguish coordinate convention from physical invariant.
A.9 Conclusion
The UPL hierarchy survives the removal of human units. In -normalized natural units, the law
collapses to , and every scale — electron, proton, neutron, human, Earth–Moon–
Sun, galaxy — converges to . The human scale is not an artifact of SI; it is the
resonant node itself.
The question worth pursuing is not "why does nature produce one second?" but "why does the
dimensionless ratio recur across systems whose masses and radii differ by dozens of
orders of magnitude?" That is the structural question, and it is the one the UPL answers.
Acknowledgment
The dimensionless reformulation presented in this appendix was suggested by Volynets Evgeny
Vatslavovich in a comment on this paper when in discussion at academia.com. His stress test of
the framework — removing human units and re-expressing the UPL in natural coordinates —
clarified the distinction between the physical resonance and its human projection. We thank him
for the constructive critique.
X(Q) = log
10
(
Q
φQ
h
)
, φ =
1 + 5
2
.
h
T = π κ R /M
T ≈ 7.4 × 10
42
7.4 × 10
42